How to tidy messy corners generated by solidify modifier?
There is not any magic way to avoid the overlapping, even with a new complex Solidify Modifier (if I'm not mistaken).

When Solidify modifier was set to Complex mode with Thickness mode Fixed, you can try ...
Weld Modifier
It worked with small distances, but already here it started to change outer topology.

With higher value of Thickness ...

... you would have to apply Solidify modifier, select inner vertex and press L to select Linked and assign those to Vertex Group under Data Properties.

Than use this vertex group as a limit for modifier.

Remesh Modifier
But ... this Weld works only because of the big flat faces inside. Probably useless in your situation. You will probably more lucky with Remesh modifier &gt; Voxel type.